MOTION OF A PARTICLE 117 Therefore the equations of motion are obtained by substituting g sin a for/ in equations (1) to (3), Thus we have v = VQ + gt sin a, s = vQt + | gt2 sin a, v*= f02+ 2gssin a, for the equations of motion. PROBLEMS. 1. A number of particles slide down smooth inclined planes of equal height. Show that the time taken by each particle to reach the base is proportional to the length of the plane along which it slides. 2. Given the base of an inclined plane, find the height so that the horizontal component of the velocity acquired in descending it may be greatest possible. 3. Two particles are projected simultaneously, one up and the other down a smooth inclined plane. Find the velocities of projection if the particles pass each other at the middle of the plane. 4. Show that the time taken by a particle to slide down any chord which begins at the highest point of a vertical circle is constant and equals 'V/f where a is the radius of the circle. 5. A particle is projected down an inclined plane of length I and height h. At the same time another particle is let fall vertically from the same point. Find the velocity of the projection of the first particle if both strike the base at the same time. 6. A ship stands at a distance d from its pier. Show that the length of the chute which will make the time of sliding down it a minimum is d V2. 107. D. Motion of a Particle along a Rough Inclined Plane. — The only difference between this problem and the last one is that the reaction of the FIG. 64. plane is not normal to the surface. On account of friction the reaction R has a component along the plane. Denoting